3. PROGRAMS FOR CALCULATING TIME-DEPENDENT
3.3 Soft Soil Creep model
∆ (3.11)
σ
∆ is unknown and iteration is made to calculate ∆ . σ
However, there have been some numerical adjustments since this model was presented by Svanö (1986) and in the present implemented model the following adjustments have been made
• The time resistance, Rc, according to eq. (3.6) is adjusted to a high value if the effective stress goes below the initial effective stress.
• For each stress increment five sub increment, i.e. ∆t/5, is used instead of two as shown in Figure 3.5.
• An elastic strain is introduced for each sub increment. This elastic strain is calculated using the oedometer modulus, MR=M0, for
stresses less then the preconsolidation stress and for stresses greater then the preconsolidation stress the oedometer modulus used is calculated as MR =M0⋅σv′max /σvc′ .
3.3 Soft Soil Creep model
The Soft Soil Creep (SSC) model is a material model implemented in the Plaxis BV finite element program. This model originates from the one-dimensional creep theories presented by e.g. Buisman (1936), Suklje (1957), Bjerrum (1967) and Garlanger (1972), and has been converted to differential form to make possible an extension to a 3D-model.
Some basic characteristics of the SSC model are:
• Stress-dependent stiffness (logarithmic compression behaviour)
• Distinction between primary loading and unloading-reloading
• Secondary compression
• Memory of preconsolidation pressure
• Failure behaviour according to the Mohr-Coulomb criteria
• Modified Cam-Clay used as a reference surface (cap) 3.3.1 Soil model
The one-dimensional version of the model in SSC is based on work carried out by Stolle et al. (1997) and Vermeer et al. (1998). The total strain, see
Programs for calculating time-dependent behaviour
0 0
ln ln pc ln 1
e c c
dc ac
p c
A σ B σ C t
ε ε ε ε
σ σ τ
′ ′
= + + = ⋅ ′ + ⋅ + ⋅ + (3.12)
Where t′ = − is the effective creep time and ε is the total logarithmic t tc strain due to an increase in effective stress from σ’0 to σ’. The total strain is divided into elastic and a visco-plastic creep part, denoted by superscript e and c respectively. The visco-plastic part can be separated into two parts, one during consolidation and one after consolidation. This is denoted by the subscript dc and ac in Figure 3.7. The values σp0, σpc and σp represent the preconsolidation stress corresponding to before loading, end of
consolidation state and after a time of pure creep respectively. The
parameters are illustrated in Figure 3.6 and Figure 3.7. εcons in Figure 3.7 represent the strain at the end of consolidation for one load step.
εcons
ε
1 /ε
εconscons
ε
ε
1 /ε
Figure 3.6 Consolidation and creep behaviour in a standard oedometer test, Brinkgreve et al. (2006).
c
Figure 3.7 An idealised stress-strain curve from an oedometer test with a division of strain increments into an elastic and a creep component, modified from Brinkgreve et al. (2006).
In the SSC-model it is assumed that the total strain is divided into elastic and inelastic strains. In this formulation the inelastic part is assumed to be purely creep, εc. The SSC model also adopts the Bjerrum´s idea that the preconsolidation stress depends only on the amount of creep strain that has accumulated over time. In addition to eq. (3.12) Vermeer & Neher (1999) introduce the following expression.
0
As can be seen from eq. (3.13) the longer it is left to creep the larger σp
grows. In a conventional IL test the load is maintained for a constant period of tc+t´=τ, where τ is exactly one day. For this type of IL test a so-called normal consolidation line with σp= σ´ is obtained. By combining eq.
(3.12) and eq. (3.13) and assuming that (τc-tc) << τ the time dependency of the preconsolidation stress can be simplified as
B
Programs for calculating time-dependent behaviour
Where τ is equal to one day, see Vermeer & Neher (1999). The differential equation can then be derived as
0exp
The one-dimensional model was extended to a general three-dimensional constitutive model based on Modified Cam-Clay type ellipses, see
Vermeer et al. (1998). The well-known stress invariants for pressure p’ and deviatoric stress q are adopted, Brinkgreve et al. (2006). These stress
invariants are used to define the size of the ellipse, see Figure 3.8, as
2
In Figure 3.8 the soil parameter MCS and MMC are shown and represent the so-called ‘critical state line’ and the Mohr-Coulomb failure line and are defined as
Where the φ´ is the effective friction angle.
Figure 3.8 Diagram of peq ellipse in a p-q plane, Satibi (2009).
Figure 3.8 suggests that tensile stresses are possible but this could be prevented by using a tension cut-off option
In the SSC-model an important feature is adopted to simulate a relative step NC surface. This is done by applying relatively large values for MCS, see Figure 3.8, which could be different from the slope MMC. The MCS could be equal to MMC, but quite large values for MCS need to be used if a prediction of more realistic K0nc values is to be obtained. Using MCS values greater than MMC will lead to a relatively steep normal consolidation
surface in a p-q plane.
In the SSC-model the parameters A, B and C above are changed to the material parameters κ*, λ* and µ*. Conversion is made accordingly, Vermeer & Neher (1999)
( )
Using the new stress invariants and parameters and omitting the elastic strain in eq. (3.15) the volumetric creep strain, εvc, could be written as
* *
If eq. (3.19) is integrated for a constant stress state the change in the size of the yield surface due to creep over a period of ∆t is
* *
where τ = one day in the SSC model. This expression defines the time-dependent creep behaviour and implies that the OCR has a considerable influence on the creep rate.
Programs for calculating time-dependent behaviour